Some Aspects of Measurement Error in Linear Regression of Astronomical Data

By B. Kelly. 2007

1,046 citations recorded by the source corpus.

I describe a Bayesian method to account for measurement errors in linear regression of astronomical data. The method allows for heteroscedastic and possibly correlated measurement errors and intrinsic scatter in the regression relationship. The method is based on deriving a likelihood function for the measured data, and I focus on the case when the intrinsic distribution of the independent variables can be approximated using a mixture of Gaussian functions. I generalize the method to incorporate multiple independent variables, nondetections, and selection effects (e.g., Malmquist bias). A Gibbs sampler is described for simulating random draws from the probability distribution of the parameters, given the observed data. I use simulation to compare the method with other common estimators. The simulations illustrate that the Gaussian mixture model outperforms other common estimators and can effectively give constraints on the regression parameters, even when the measurement errors dominate the observed scatter, source detection fraction is low, or the intrinsic distribution of the independent variables is not a mixture of Gaussian functions. I conclude by using this method to fit the X-ray spectral slope as a function of Eddington ratio using a sample of 39 z ≲ 0.8 radio-quiet quasars. I confirm the correlation seen by other authors between the radio-quiet quasar X-ray spectral slope and the Eddington ratio, where the X-ray spectral slope softens as the Eddington ratio increases. IDL routines are made available for performing the regression.

Subfields: Artificial Intelligence, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/957fb2696feed2ebba263c0f4a453c346fde2733

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Citations in the corpus

A heteroscedastic measurement error model based on skew and heavy-tailed distributions with known error variances (2018, background): “To deal with such data, heteroscedastic MEMs with known error variances have been studied in areas such as Medicine [6,7], Analytical Chemistry [8] and Astrophysics [9], among others.”
REVISITING THE SCALING RELATIONS OF BLACK HOLE MASSES AND HOST GALAXY PROPERTIES (2012, neutral): “LINMIX ERR is a Bayesian estimator by Kelly (2007).”
The subarcsecond mid-infrared view of local active galactic nuclei – II. The mid-infrared–X-ray correlation (2015, methodology): “To measure the functional description of the correlations, we perform linear regression in logarithmic space with the Bayesian based linmix err algorithm and a normal prior distribution (Kelly 2007).”
Cold Molecular Outflows in the Local Universe and Their Feedback Effect on Galaxies (2018, methodology): “This and the following fits are performed by using linmix (Kelly 2007), considering the error bars both in x and y and including upper limits.”
COMPARING SINGLE-EPOCH VIRIAL BLACK HOLE MASS ESTIMATORS FOR LUMINOUS QUASARS (2012, methodology): “The green lines are the linear regression results using the Bayesian method of Kelly (2007).”
The ATLAS3D project – XX. Mass–size and mass–σ distributions of early-type galaxies: bulge fraction drives kinematics, mass-to-light ratio, molecular gas fraction and stellar initial mass function (2012, methodology): “When the same outliers are removed, we verified that indistinguishable values for the parameter, their errors and the inferred intrinsic scatter are obtained with the Bayesian approach of Kelly (2007), as implemented in his IDL routine LINMIX_ERR.”
Clarifying the Hubble constant tension with a Bayesian hierarchical model of the local distance ladder (2017, methodology): “Fitting the SN population using a BHM removes the bias injected by the χ2minimization routines popular in the literature (see Kelly 2007; March et al. 2011; Mandel et al. 2016 for discussion).”
ROBUST OPTICAL RICHNESS ESTIMATION WITH REDUCED SCATTER (2011, methodology): “The fit is done following the algorithm presented in Kelly (2007), which performs a full Bayesian modeling of the powerlaw distribution, and correctly takes into account errors on the independent variable as well as upper limits on LX for those clusters without significant detection of X-ray…”
Planck2018 results (2018, methodology): “The slopes of the correlations are obtained using the Bayesian fitting method of Kelly (2007). The reduced χ2, the Pearson correlation coefficient, and the correlation coefficient inferred from the Bayesian method (Kelly 2007) are listed.”
TYPE Iax SUPERNOVAE: A NEW CLASS OF STELLAR EXPLOSION (2012, methodology): “In other words, a SN that is bright and declines slowly in B is also bright and declines slowly in R. Performing a Bayesian Monte-Carlo linear regression on the data (Kelly 2007), we determine correlations between different parameters in different bands.”
An ALMA Survey of Protoplanetary Disks in the σ Orionis Cluster (2017, methodology): “The procedure most often utilized in the disk survey literature is the Bayesian linear regression method of Kelly (2007), as it is capable of accounting for these three key factors simultaneously, unlike other linear regression methods (see Pascucci et al. 2016 for a detailed discussion).”
THE SINFONI BLACK HOLE SURVEY: THE BLACK HOLE FUNDAMENTAL PLANE REVISITED AND THE PATHS OF (CO)EVOLUTION OF SUPERMASSIVE BLACK HOLES AND BULGES (2016, methodology): “We compute ZP , ~α and following Kelly (2007) and making use of his IDL routines.”
PROBABILISTIC MASS–RADIUS RELATIONSHIP FOR SUB-NEPTUNE-SIZED PLANETS (2015, methodology): “A very similar approach to this HBMenabled linear regression was detailed in Kelly (2007); we refer the reader to that paper for an in-depth discussion of the general advantages and improvements of this approach over the commonly used χ2 analysis for linear regression.”
Cosmological Parameters from Observations of Galaxy Clusters (2011, background): “The distribution of the relation’s independent variable(s) (in this case cluster masses) within the full population generically influences constraints on scaling laws (e.g. Gelman et al. 2004; Kelly 2007).”
SiFTO: An Empirical Method for Fitting SN Ia Light Curves (2008, methodology): “We use the multiple regression method of Kelly (2007), which also provides estimates of σclr, and couple these fits with iterative 2.5σ outlier rejection when deriving the relation.”
Robust weak-lensing mass calibration of Planck galaxy clusters (2014, methodology): “However, we choose to quote the more conservative error estimates returned by the Kelly (2007) method.”
The SDSS–2MASS–WISE 10-dimensional stellar colour locus (2014, methodology): “A linear regression for each of the Aλ versus Ar distributions was computed, weighting each data point by the photometric error, using a Bayesian approach of sampling from the posterior distribution with the IDL routine LINMIX ERR (Kelly 2007).”